This past week my 7th grade class was starting on multiplying rational numbers….which of course means...it’s time to see what they remember from 6th grade. I went back and forth on how to present the review...do I just remind them of the algorithm? Do I do a quick “hey, remember this model” then jump to the algorithm? Do I have them reinvent the algorithm? Due to the fact that I knew they modeled multiplying fractions last year and that we’ve been having some hard core discussions about what our operations mean, I determined that I would just see what they could do.

I had students working in groups of 3-4 and in the middle of the table was a plethora of manipulatives...a sampling of pretty much everything I have in my cabinets. I gave each group a note card with a fraction multiplication problem and the following prompt: "Use the stuff at your table to model your multiplication problem as many different ways as you can."

I'll be honest, I had pretty high expectations of what I'd get. We've been doing What's Your Story every week since the start of school and we've had many conversations about what a fraction of a group could look like. Unfortunately, this is what most groups did:

I decided to take a step back and discuss as a class what 1/2 x 1/2 means...1/2 of a group of 1/2. I asked each group to show a way to model it. Things went a little better, but I'll be honest...after the first flop, I was feeling defeated...I just wasn't as excited as I was at the beginning of the lesson. I was having that initial internal struggle with myself:

Do I spend time solidifying what multiplying fractions means before adding in decimals and negatives?

Do I say "screw it" and just remind them, "Hey! Remember that you just multiply across? Yes, the easiest thing you ever have to do with fractions can be completely convoluted and confusing by using these models I'm trying to force on you in order to glean some understanding of what's happening to the numbers???"

How soon do I start throwing in mixed numbers?

It's almost November and I'm STILL on Unit 1! WTF am I doing???

Long story short, I decide to put the manipulatives away and try our hand at modeling on paper using grids. We did some problems together and with about 5 minutes left in class I gave them a ticket out the door. Results were pretty much what I expected. About 1/3 of them tried to model and failed, 1/3 were able to draw the model and 1/3 didn't even try the model and just multiplied across.

I'd like to say that after that first failure I was able to change things up for the other classes in a way that made things wonderful for them and me...unfortunately, even with the minor changes I made to the lesson (like starting with the 1/2 x 1/2 conversation) the results were pretty much the same. My lower students were getting confused with the model. The higher students were annoyed that I was even asking them to model. It was just, overall, not a great math day.

Ideally, this is the part of the blog where I'd talk about what I'd do differently...how I'd adjust my lesson for next year...but at this point my brain isn't ready to process that. Right now I'm trying to think about what went so wrong...what did I do differently from years past to make this lesson flop so hard. If I ever figure it out, I'll let you know.

Today we reviewed converting fractions into decimals using the following worksheet. (Thanks to Illustrative Mathematics for the last two questions.)

While walking around, I saw a pair of students who said 1/2 was equal to 1.2. This is a pretty common error in 7th grade. Here's a summary of the conversation we had (Eventually I'll remember to use my voice recorder on my phone for better accuracy.):

Me: Can you show me where 1/2 is on a number line?Student 1: It's half way between 0 and 1.Me: Where would 1.2 be on a number line?Student 2: It's after the 1. Between the 1 and the 2.Me: So, can 1.2 be equal to 1/2?

(At this point, there was a bit of discussion between the girls about whether or not they placed the numbers on the number line correctly. After they decided they were correctly placed, they determined that 1/2 could not equal 1.2.)

S2: Oh yeah, if it was 1.2, there would be a 1 in front of the fraction, so it has to be 0 point something.Me: So, what do you think 1/2 is as a decimal?S1: 0.2 Me: Why?S2: Because it has to be a 0 before the decimal and there's a 2.

All right...back to the number line...

Me: Can you show me what that would look like on the number line?S2: If I split this in to 4 equal pieces, 1/2 is at the second line, so that would be 0.2

Me: Ok (I point to the other marks.) Can you tell me what these would be, then?S2: (Points to the 1st mark) This would be 0.1. (Points to the 3rd mark) This would be 0.3. (Points to the 4th mark) This would be 0.4

​(At this point, I've basically sat back and let the girls go...hoping for a glimmer of light.)

S1: That's not right.​S2: Yeah, 0.4 is not 1.....Wait....we need to split this in to 10 pieces.

(Oh, thank goodness...)

Me: Can you explain why?S2: Because that's what the decimal place means. That's the tenths place.​Me: Ok, so try that.

S1: So, it's 0.5?Me: Does that make sense?S2: Yeah, because it's like 50 cents is half of a dollar.​S1: Oh yeah.Me: Awesome.

We had a few more struggles along the way, but after that first interaction, their desire to just replace the fraction line with the decimal stopped. They actually starting thinking about what the denominator meant in relation to the decimal equivalence.